Thin lenses — lens equation, lens maker's equation: Visual walkthrough

We are chasing one target: where does a lens focus light, and why does its shape decide that?

Step 1 — What a ray, a normal, and an angle actually are

WHAT. Before any formula, we agree on the cast of characters in the picture below.

A ray is the straight path light travels (the arrow).
A spherical surface is the boundary of the glass, curved like part of a ball. The centre of that ball is the point $C$ , called the centre of curvature. The distance from the surface to $C$ is the radius $R$ .
The normal is the line straight out of the surface — for a sphere it always points along the line to $C$ . This matters because angles in optics are always measured from the normal, never from the glass.
The optical axis is the straight horizontal centre line; everything is measured along it.

WHY. Light bends according to the angle it makes with the normal. If we don't know where the normal points, we can't measure the angle, and nothing else follows. On a curved surface the normal tilts as you move up the glass — that tilt is the whole reason a lens focuses.

PICTURE.

Figure — Thin lenses — lens equation, lens maker's equation

Step 2 — Snell's law, in the only form we need

WHAT. When light crosses from one material into another, it bends. Each material has an index $n$ : how many times slower light travels in it than in vacuum. Air is $n \approx 1$ ; glass is $n \approx 1.5$ .

The full law is $n_{1} sin i = n_{2} sin r$ , where $i$ is the angle in from the normal and $r$ the angle out.

WHY this tool and not another? We need a rule that connects the incoming tilt to the outgoing tilt at a surface — that is exactly refraction, and Snell's law is the one statement of it. We use the paraxial form because our angles are tiny:

$n_{1} i = n_{2} r$

Term by term: $n_{1}$ = index the ray leaves; $i$ = angle it made with the normal on the way in; $n_{2}$ = index it enters; $r$ = angle with the normal on the way out. Bigger $n_{2}$ forces a smaller $r$ — the ray bends toward the normal entering denser glass.

PICTURE.

Step 3 — Three little triangles give three angles

WHAT. Take one ray from an on-axis object point $O$ (far left) that strikes the surface at height $h$ . It refracts and heads toward the image point $I$ . We name three angles the ray-lines make with the axis:

$α$ — angle of the incoming ray at the object,
$β$ — angle of the outgoing ray at the image,
$ϕ$ — angle the normal (the line to $C$ ) makes with the axis.

Using paraxial " $tan θ \approx height / distance$ ", and reading distances with the sign convention ( $u < 0$ for a real object on the left, $v > 0$ , $R > 0$ if $C$ is on the outgoing side):

$α = \frac{h}{- u}, β = \frac{h}{v}, ϕ = \frac{h}{R}$

Term by term: each is the same height $h$ divided by how far away the relevant point ( $O$ , $I$ , $C$ ) sits. The $- u$ appears because $u$ is negative but a physical distance is positive.

WHY. We can't plug "angles" into Snell's law until we express them in things we can measure: heights and distances. These three ratios convert geometry into the numbers $u, v, R$ that we ultimately want.

PICTURE.

Step 4 — The exterior-angle trick turns angles into $i$ and $r$

WHAT. An exterior angle of a triangle equals the sum of the two opposite interior angles. Apply it twice to the picture:

$i = α + ϕ and r = ϕ - β$

WHY. Snell's law is written in $i$ and $r$ (measured from the normal), but our measurable ratios in Step 3 are $α, β, ϕ$ (measured from the axis). This trick is the bridge: it re-expresses the normal-angles using axis-angles.

Reading it: for $i$ , the incoming ray-to-axis angle $α$ plus the normal's own tilt $ϕ$ stacks up to the full angle from the normal. For $r$ , the ray has swung past, so we subtract $β$ .

PICTURE.

Step 5 — Substitute, cancel $h$ , and the single-surface law appears

WHAT. Feed Step 4 into paraxial Snell $n_{1} i = n_{2} r$ , then insert Step 3's ratios:

$n_{1} (\frac{h}{- u} + \frac{h}{R}) = n_{2} (\frac{h}{R} - \frac{h}{v})$

Every term carries the same height $h$ , so divide it out (this is why the answer doesn't depend on which ray we picked — all paraxial rays agree, which is why an image forms at all):

$- \frac{n _{1}}{u} + \frac{n _{1}}{R} = \frac{n _{2}}{R} - \frac{n _{2}}{v}$

Rearrange to collect image terms on the left:

$\frac{n _{2}}{v} - \frac{n _{1}}{u} = \frac{n _{2} - n _{1}}{R}$

Term by term: $\frac{n _{2}}{v}$ = "where the image is, weighted by the glass it sits in"; $\frac{n _{1}}{u}$ = same for the object; the right side $\frac{n _{2} - n _{1}}{R}$ = the surface's own bending strength — big when the media differ a lot or the curve is tight.

WHY. This is the single-surface formula — our reusable building block. One curved surface, done forever.

PICTURE.

Step 6 — Do it twice: a lens is two surfaces back to back

WHAT. A thin lens is glass ( $n$ ) in air. Light meets surface 1 (radius $R_{1}$ , air→glass), then surface 2 (radius $R_{2}$ , glass→air). "Thin" means the two surfaces sit at the same plane — no gap.

Surface 1, with $n_{1} = 1$ , $n_{2} = n$ , the image landing at an intermediate point $v_{1}$ :

$\frac{n}{v _{1}} - \frac{1}{u} = \frac{n - 1}{R _{1}} (1)$

The intermediate image of surface 1 is the object for surface 2. Because the lens is thin, that object also sits at distance $v_{1}$ . Now $n_{1} = n$ , $n_{2} = 1$ :

$\frac{1}{v} - \frac{n}{v _{1}} = \frac{1 - n}{R _{2}} (2)$

WHY. We never re-derive anything — we reuse Step 5's box, once per surface. That is the entire economy of the approach.

PICTURE.

Step 7 — Add the two equations: the ghost image cancels

WHAT. Add (1) and (2). The intermediate term appears as $+ \frac{n}{v _{1}}$ in the first and $- \frac{n}{v _{1}}$ in the second — they destroy each other:

$\frac{1}{v} - \frac{1}{u} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

Now define the focal length $f$ as where parallel incoming rays (object infinitely far, $u \to - \infty$ , so $1/ u \to 0$ ) converge, i.e. $1/ v = 1/ f$ :

$\frac{1}{f} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) Lens maker’s equation$

and, comparing the two boxed lines, the thin-lens equation:

$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

WHY. The cancellation is the miracle: we never needed to know where the ghost image $v_{1}$ was. Two surfaces collapse into one number $f$ that depends only on the shape and glass.

Term by term in the maker's equation: $(n - 1)$ = "how much the glass slows light" (if $n = 1$ it's zero, no lens at all); $(\frac{1}{R _{1}} - \frac{1}{R _{2}})$ = "total curvature difference of the two faces".

PICTURE.

Step 8 — The degenerate & edge cases you must never trip on

WHAT. Check every limit so no scenario surprises you.

Case	Input	Result	Reading
Flat plate	$R_{1}, R_{2} \to \infty$	$1/ f = 0$	Zero curvature → does nothing
Symmetric biconvex	$R_{1} = + R, R_{2} = - R$	$\frac{1}{f} = (n - 1) \frac{2}{R}$	Signs make faces add
Diverging (biconcave)	$R_{1} < 0, R_{2} > 0$	$f < 0$	Spreads rays, virtual focus
No glass	$n = 1$	$1/ f = 0$	$(n - 1) = 0$ : no speed change, no bend
Object at $f$	$u = - f$	$v \to \infty$	Rays leave parallel

WHY. A formula you trust is one you've watched behave correctly at its extremes. The two most dangerous sign slips are (a) forgetting $R_{2}$ is negative for a biconvex lens, and (b) writing a "+" like the mirror equation.

PICTURE.

The one-picture summary

Recall Feynman retelling — the whole walkthrough in plain words

Picture one ray leaving a dot on the left, hitting a curved bit of glass. To know how much it bends I need the "straight-out" line at that spot (the normal) — for a ball, that line always points at the ball's centre. I compare the ray to that line: small angle in, small angle out, and glass makes it swing (Snell). I turn all those angles into simple "height ÷ distance" fractions, plug them in, and the height cancels — which is why all the rays from one dot meet at one new dot: the image. That gives me one rule for one curved surface. A lens is just two surfaces kissing, so I use that rule twice — the leftover "middle image" appears with a plus in one line and a minus in the other and vanishes when I add them. What survives is a single number $f$ built from the glass ( $n - 1$ ) and the two curves. Flat glass? No curve, no focus. Fatter, tighter curves, denser glass? Stronger swing, closer focus. That's the whole story.

Recall

Why does the height $h$ cancel in Step 5? ::: Every angle was written as $h / distance$ , so $h$ is a common factor — dividing it out means all paraxial rays obey the same relation, so they focus to one point. Why does the intermediate image $v_{1}$ disappear? ::: It enters as $+ n / v_{1}$ and $- n / v_{1}$ in the two surface equations; adding them cancels it exactly. What defines the focal length $f$ in the derivation? ::: The image distance when the object is infinitely far ( $u \to - \infty$ , $1/ u \to 0$ ), so $1/ v = 1/ f$ . For a symmetric biconvex lens, why do the faces "add"? ::: $R_{1} = + R$ and $R_{2} = - R$ , so the subtraction $1/ R_{1} - 1/ R_{2}$ becomes $2/ R$ .

Connections

Thin lenses — lens equation, lens maker's equation (parent)
Refraction at a single spherical surface
Snell's law & paraxial approximation
Spherical mirrors — mirror equation
Magnification and image formation
Power of a lens (dioptres)
Lens combinations & equivalent focal length
Lens aberrations (chromatic, spherical)